Method for compressing digital images

ABSTRACT

The invention concerns a method for compressing data, in particular images, by transform, in which method this data is projected onto a base of localized orthogonal or biorthogonal functions, such as wavelets. To quantize each of the localized functions with a quantization step that enables an overall set rate R c  to be satisfied, the method includes the following steps:  
     a probability density model of coefficients in the form of a generalized Gaussian is associated with each subband,  
     the parameters α and β of this density model are estimated, while minimizing the relative entropy, or Kullback-Leibler distance, between this model and the empirical distribution of coefficients of each subband, and  
     from this model, for each subband, an optimum quantization step is determined such that the rate allocated is distributed in the various subbands and such that the total distortion is minimal. As a preference, for each subband, the graphs of rate R and distortion D are deduced, from the parameters α and β, as a function of the quantization step and these graphs are tabulated to determine said optimum quantization step.

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application is based on French Patent Application No. 02 05724 filed May 7, 2002, the disclosure of which is hereby incorporated byreference thereto in its entirety, and the priority of which is herebyclaimed under 35 U.S.C. §119.

BACKGROUND OF THE INVENTION

[0002] 1. Field of the Invention

[0003] The invention relates to a method for optimal allocation of thebit rate of a data compressor by transform.

[0004] It concerns more specifically such an allocation process for acompressor by orthogonal or biorthogonal transformation, in particular awavelet transformation, used in combination with a scalar quantizer anda lossless entropy coder.

[0005] Hereinafter, reference will mainly be made to a wavelettransformation leading to a Multiresolution Analysis (MRA) implementedusing digital filters intended to perform a decomposition into subbands.However, the invention is not limited to a wavelet transformation.

[0006] It is recalled that an MRA consists in starting from an image inthe space domain with a set of image elements, or pixels, and indecomposing this image into subbands in which the vertical, horizontaland diagonal details are represented. Thus there are three subbands perresolution level as indicated in FIG. 1.

[0007]FIG. 1 illustrates an MRA on three resolution levels. In thisrepresentation, a subband is represented by a block. Thus, the image isfirst divided into four blocks with three subbands W_(1,1), W_(1,2) andW_(1,3) and a low-frequency representation W_(1,0) of the initial image.Subband W_(1,1) contains the horizontal wavelet coefficients; subbandW_(1,2) contains the vertical wavelet coefficients; subband W_(1,3)contains the diagonal wavelet coefficients and the block W_(1,0) iscalled “summary” or low frequencies.

[0008] At the next resolution level, the block W_(1,0) is itself dividedinto four blocks (one summary and three subbands) W_(2,0,) W_(2,1),W_(2,2) and W_(2,3) and, finally, the block W_(2,0) is divided into fourblocks W_(3,0), W_(3,1), W_(3,2) and W_(3,3) for the third Naturally, afiner division (by increasing the resolution levels) or a coarserdivision (by reducing the number of resolution levels) can be carriedout.

[0009] 2. Description of the Prior Art

[0010] It is known that the wavelet transform is well suited to imagecompression since it provides strong coefficients when the imageexhibits strong local variations in contrast, and weak coefficients inthe areas in which the contrast varies slightly or slowly.

[0011] It is also known that the probability distribution of a subbandcan be modeled by a two-parameter unimodal function, centered at theorigin, of the generalized Gaussian type:${G_{\alpha\beta}(x)} = {\frac{\beta}{2\quad {{\alpha\Gamma}\left( \frac{1}{\beta} \right)}}^{- {\frac{x}{\alpha}}^{\beta}}}$

[0012] where Γ(ξ) = ∫₀^(+∞)^(−x)x^(ξ − 1)  x

[0013] For certain applications, particularly when the compression datamust be transmitted over transmission channels imposing a bit rate, itis necessary to quantize the subband coefficients in an optimal mannerby minimizing the total distortion while satisfying a set bit rate.

[0014] The known optimum rate allocation methods propose, in general,performing a digital optimization process based on the minimization of afunctional linking rate and distortion and controlled by a Lagrangeparameter. In this case, an iterative optimization algorithm is employedwhich is generally very costly in calculation time and therefore cannotbe used for real-time applications or for applications with limitedcalculation resources. Only simplification of these methods enableshigher speed, but the price is a degradation in performance.

SUMMARY OF THE INVENTION

[0015] The invention enables an effective and precise optimization ofrate allocation, without making use of a conventional Lagrangianoptimization scheme.

[0016] To implement an optimum rate allocation, the method in accordancewith the invention includes the following steps:

[0017] a) the image (or data) to be compressed is projected onto a baseof localized orthogonal or biorthogonal functions, such as wavelets,

[0018] b) a set rate R_(c) is chosen, representing the total number ofbits that can be used to code all coefficients of the transformed image,

[0019] c) to allocate this number of bits to the coefficients which aregoing to be quantized:

[0020] c.1 a probability density model in the form of a generalizedGaussian is associated with each of the subbands,

[0021] c.2 the parameters of this generalized Gaussian are estimatedwhile minimizing the relative entropy, or Kullback-Leibler distance,between this generalized Gaussian and the empirical distribution ofcoefficients of each subband, and an optimum quantization step is thendeduced therefrom, which is such that the total allocated rate R_(c) isdistributed in the various subbands while minimizing the totaldistortion.

[0022] Generalized Gaussians are described in the article by S. MALLAT:“Theory for multiresolution signal decomposition: the waveletrepresentation”, published in IEEE Transaction on pattern analysis andmachine intelligence, vol. 11 (1989) No. 7, pages 674-693.

[0023] As a preference, to determine the optimum quantization step foreach subband, from the parameters of the generalized Gaussian, the graphof rate as a function of the quantization step and the graph ofdistortion as a function of the quantization step are determined, andthe graphs of rate and distortion are tabulated and, from thesetabulated graphs, the optimum quantization step is deduced.

[0024] Minimization of the Kullback-Leibler distance for estimating theparameters of the generalized Gaussian model ensures a minimization ofthe cost of coding in accordance with information theory.

[0025] Thus, the invention concerns a method for compressing data, inparticular images, in which method this data is projected onto a base oflocalized orthogonal or biorthogonal functions, and which method, toquantize each of the localized functions with a quantization step thatenables an overall set rate R_(c) to be satisfied, includes thefollowing steps:

[0026] a) a probability density model of coefficients in the form of ageneralized Gaussian is associated with each subband,

[0027] b) the parameters α and β of this density model are estimatedwhile minimizing the relative entropy, or Kullback-Leibler distance,between this model and the empirical distribution of coefficients ofeach subband, and

[0028] c) from this model, for each subband, an optimum quantizationstep is determined such that the rate allocated is distributed in thevarious subbands and such that the total distortion is minimal.

[0029] As a preference, for each subband, the graphs of rate R anddistortion D are deduced, from the parameters α and β, as a function ofthe quantization step and these graphs are tabulated to determine saidoptimum quantization step.

[0030] The transformation is, for example, of the wavelet type.

[0031] In this case, according to an embodiment, to determine theparameter β of the generalized Gaussian associated with each subband,the following expression is minimized:${{H\left( p_{1}||G_{{\alpha {(\beta)}},\beta} \right)} = {\frac{1}{\beta} + {\frac{1}{\beta}{\log \left( {\beta \quad W_{jk}^{\beta}} \right)}} + {\log \left( \frac{2{\Gamma \left( {1/\beta} \right)}}{\beta} \right)}}},$

[0032] in which formula the first term represents said relative entropy,W_(jk)^(β)

[0033] has the value:$W_{jk}^{\beta} = {\frac{1}{n_{jk}}{\sum\limits_{n,m}^{\quad}\quad {{W_{jk}\left\lbrack {n,m} \right\rbrack}}^{\beta}}}$

[0034] W_(jk)[n,m] being a coefficient of a subband, and n_(jk) beingthe number of coefficients in the subband of index j,k.

[0035] As a preference, the parameter ox of the generalized Gaussian forthe corresponding subband j,k is determined by the following formula:$\alpha = \sqrt[\beta]{\beta \quad W_{jk}^{\beta}}$

[0036] The tabulation of the distortion values is advantageously carriedout for a sequence of values of the parameter β.

[0037] Similarly, the tabulation of the rates R is advantageouslycarried out for a sequence of values of the parameter β.

[0038] The sequence of values of the parameters β is for example:$\left\{ {\frac{1}{2},\frac{1}{\sqrt{2}},1,{\sqrt{2,}2}} \right\}$

[0039] As a preference, the bit budget is distributed to each of thesubbands according to their ability to reduce the distortion of thecompressed image.

[0040] In one embodiment, a bit budget corresponding to the highesttabulated quantization step is assigned to each subband and theremaining bit budget is then cut into individual parts that aregradually allocated to the localized functions having the greatestability to make the total distortion decrease, this operation beingrepeated until the bit budget is exhausted.

[0041] According to one embodiment, the localized functions and thequantization step for each subband and the parameters of the densitymodel are coded using a lossless entropy coder.

[0042] Other features and advantages of the invention will becomeapparent with the description of some of its embodiments, thisdescription being provided with reference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

[0043]FIG. 1, already described, shows the decomposition into subbandsof an image on three resolution levels.

[0044]FIG. 2 is a diagram showing graphs of rate for an exampleapplication of the method according to the invention.

[0045]FIG. 3 is a diagram similar to that of FIG. 2 for graphs ofdistortion.

[0046]FIG. 4 is a schematic diagram of a device implementing the methodaccording to the invention to which a control means has been added,which is intended to manage the filling of a buffer memory the role ofwhich is to deliver a constant number of bits to a transmission channelor to any other device requiring a fixed rate.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0047] The example of the invention that will be described belowconcerns an allocation of rate by subbands. It is carried out in threesteps:

[0048] First Step: Wavelet Transformation of the Image

[0049] This transformation provides a family of wavelet coefficientsW_(jk)[n,m] distributed in various subbands, where j denotes the scaleof the subband and k its orientation (FIG. 1). The number of indexes n,mfor which wavelet coefficients are thus defined depends on the scale ofthe subband. It is therefore noted n_(jk).

[0050] Consider, for example, an image of size 512×512 pixels of aterrestrial landscape obtained by an observation satellite. The size ofa subband j,k is in this case n_(jk)=(512/2^(j))².

[0051] Second Step: A total rate setting is allocated for all thesubbands and this total rate setting is distributed among the varioussubbands.

[0052] To assign the rate in each subband, the procedure is as follows:

[0053] a) It is considered that the coefficients in each subband j,k aredistributed according to a statistical distribution corresponding to ageneralized Gaussian of parameters α and β. These parameters α and β areestimated, in a manner that will be described later, while minimizingthe Kullback-Leibler distance between this statistical model and theempirical distribution of coefficients of each subband.

[0054] b) The knowledge of these parameters α and β can be used topredict, for each subband j,k, the distortion D_(jk) which will beassociated with a rate R_(jk) which will be allocated to it, and therelationship between these two parameters α and β and the quantizationstep Δ_(jk) chosen for this subband. Rate scheduling is carried out bydistributing the rate setting R among the subbands j,k, in such a waythat the complete rate is allocated to them:$R = {\sum\limits_{jk}^{\quad}R_{jk}}$

[0055] and that the resulting distortion$D = {\sum\limits_{jk}^{\quad}D_{jk}}$

[0056] is minimal.

[0057] At the end of the scheduling phase, an optimum quantization stepΔ_(jk) has therefore been determined for each subband, enabling the ratesetting R to be reached while minimizing the total distortion D.

[0058] Third step: The subbands are quantized with quantization stepsΔ_(jk) which can be different from one subband to another and thequantized coefficients are sent to an entropy coder.

[0059] Determination of Parameters α and β of the Generalized Gaussiansfor each Subband:

[0060] For each subband j,k, the statistical distribution of the subbandtends toward a generalized Gaussian G_(αβ)z:${G_{\alpha\beta}(x)} = {A_{\alpha\beta}^{- {\frac{x}{\alpha}}^{\beta}}}$

[0061] where$A_{\alpha\beta} = \frac{\beta}{2{{\alpha\Gamma}\left( {1/\beta} \right)}}$

[0062] To determine the parameters αand β, as indicated above, therelative entropy between this subband density model and the empiricaldensity of this subband is minimized.

[0063] It is recalled here that the relative entropy, orKullback-Leibler distance, between two probability densities p₁ and p₂is given by:${D\left( p_{1}||p_{2} \right)} = {\int{{p_{1}(x)}\log \frac{p_{1}(x)}{p_{2}(x)}{{x}.}}}$

[0064] In the sense of this Kullback-Leibler distance, the distributionp₂ which best tends toward the distribution p₁ is that which minimizesD(p₁∥p₂).

[0065] Note furthermore that to determine p₂ which minimizes D(p₁∥p₂),for a fixed p₁, the following is minimized:

D(p ₁ ∥p ₂)=∫p ₁(x)log p ₁(x)dx−∫p ₁(x)log p ₂(x)dx.

[0066] As the first term of this difference does not depend on p₂,minimizing this sum in P₂ amounts to minimizing the second term of thissum, and therefore amounts to minimizing:

H(p ₁ ∥p ₂)=−∫p₁(x)log p ₂(x)dx   (1)

[0067] If p₁ and p₂ are discrete distributions, the term above is thenthe average rate of coding of a source of symbols of probabilitydistribution p₁, coded with optimum entropy symbols for a distributionp₂.

[0068] Minimizing this term amounts therefore to choosing a modeldistribution p₂ which will produce the most efficient symbols for codinga distribution source p₁.

[0069] Therefore D(p₁∥p₂) will be minimized and not the reverse,D(p₂∥p₁), since the Kullback-Leibler distance is not symmetric.

[0070] In the present case, the distribution p₁ is the empiricaldistribution of a subband (j,k): $\begin{matrix}{{p_{1}(x)} = {\frac{1}{n_{jk}}{\sum\limits_{n,m}^{\quad}\quad {\delta \left( {x - {W_{jk}\left\lbrack {n,m} \right\rbrack}} \right)}}}} & (2)\end{matrix}$

[0071] and the distribution p₂ is the generalized Gaussian indicatedabove: $\begin{matrix}{{p_{2}(x)} = {{G_{\alpha\beta}(x)} = {A_{\alpha\beta}^{- {\frac{x}{\alpha}}^{\beta}}}}} & (3)\end{matrix}$

[0072] The expression (1) calculated with (2) and (3) gives:$\begin{matrix}{{H\left( p_{1}||G_{\alpha\beta} \right)} = {{- \frac{1}{n_{jk}}}{\sum\limits_{n,m}^{\quad}{\log \left( {A_{\alpha\beta}^{- {\frac{W_{jk}{\lbrack{n,m}\rbrack}}{\alpha}}^{\beta}}} \right)}}}} \\{= {{{- \log}\quad A_{\alpha\beta}} + {\frac{1}{n_{jk}}{\sum\limits_{n,m}^{\quad}{\frac{W_{jk}\left\lbrack {n,m} \right\rbrack}{\alpha}}^{\beta}}}}}\end{matrix}$

[0073] Using the value of A_(αβ) gives: $\begin{matrix}{{H\left( p_{1}||G_{\alpha\beta} \right)} = {{{- \log}\quad \beta} + {\log \quad 2} + {\log \quad \alpha} + {\log \quad {\Gamma \left( {1/\beta} \right)}} + {\frac{1}{n_{jk}}{\sum\limits_{n,m}^{\quad}\quad {\frac{W_{jk}\left\lbrack {n,m} \right\rbrack}{\alpha}}^{\beta}}}}} & (4)\end{matrix}$

[0074] Minimization in α,β is carried out in two steps: during the firststep, α is minimized for a fixed β; this minimization is performed by asimple explicit calculation. During the second step, a search is carriedout for an optimum β using tabulated values to avoid calculations thatare too complex.

[0075] Minimization of α for a Fixed β

[0076] To minimize expression (4) in α, with a fixed β, the sum of theterms that depend on α must be minimized, other terms being constants,that is:${\log \quad \alpha} + {\frac{1}{n_{jk}}{\sum\limits_{n,m}^{\quad}{\frac{W_{jk}\left\lbrack {n,m} \right\rbrack}{\alpha}}^{\beta}}}$

[0077] The derivative of this expression with respect to α is:$\frac{1}{\alpha} - {\frac{\beta}{\alpha^{\beta + 1}n_{jk}}{\sum\limits_{n,m}^{\quad}{\frac{W_{jk}\left\lbrack {n,m} \right\rbrack}{\alpha}}^{\beta}}}$

[0078] and its derivative is cancelled for$\alpha = \sqrt[\beta]{\frac{\beta \quad}{n_{jk}}{\sum\limits_{n,m}^{\quad}{\frac{W_{jk}\left\lbrack {n,m} \right\rbrack}{\alpha}}^{\beta}}}$

[0079] To lighten the notations, the moment of order β of the subbandj,k is defined:$W_{jk}^{\beta} = {\frac{1}{n_{jk}}{\sum\limits_{n,m}^{\quad}{{W_{jk}\left\lbrack {n,m} \right\rbrack}}^{\beta}}}$

[0080] which is therefore the average of the absolute values ofcoefficients of the subband, raised to the power β.

[0081] The optimum α is then written simply: $\begin{matrix}{\alpha = \sqrt[\beta]{\beta \quad W_{jk}^{\beta}}} & (5)\end{matrix}$

[0082] Calculation of Optimum β

[0083] It is therefore known how to determine the optimum α once β isknown. To determine the optimum β, in equation (4) α is replaced by thevalue given by equation (5). The following is obtained: $\begin{matrix}{{H\left( p_{1}||G_{a\beta} \right)} = {{{- \log}\quad \beta} + {\log \quad 2} + {\frac{1}{\beta}{\log \left( {\beta \quad W_{jk}^{\beta}} \right)}} + {\log \quad {\Gamma \left( {1/\beta} \right)}} + {\frac{1}{n_{jk}}{\sum\limits_{n,m}^{\quad}\quad {\frac{W_{jk}\left\lbrack {n,m} \right\rbrack}{\sqrt[\beta]{\beta \quad W_{jk}^{\beta}}}}^{\beta}}}}} \\{= {{{- \log}\quad \beta} + {\log \quad 2} + {\frac{1}{\beta}{\log \left( {\beta \quad W_{jk}^{\beta}} \right)}} + {\log \quad {\Gamma \left( {1/\beta} \right)}} + \frac{W_{jk}^{\beta}}{\beta \quad W_{jk}^{\beta}}}} \\{= {{{- \log}\quad \beta} + {\log \quad 2} + {\frac{1}{\beta}{\log \left( {\beta \quad W_{jk}^{\beta}} \right)}} + {\log \quad {\Gamma \left( {1/\beta} \right)}} + \frac{1}{\beta}}}\end{matrix}$

[0084] The optimum value of β will therefore be obtained by minimizingthe expression above which can again be rewritten in the following form:$\begin{matrix}{{H\left( {p_{1}{G_{a\beta}}} \right)} = {\frac{1}{\beta} + {\frac{1}{\beta}{\log \left( {\beta \quad W_{jk}^{\beta}} \right)}{\_ log}\left( \frac{2{\Gamma \left( {1/\beta} \right)}}{\beta} \right)}}} & (6)\end{matrix}$

[0085] Thus, the calculation of the optimum α,β pair is carried out intwo stages:

[0086] First, β is calculated by minimizing expression (6).

[0087] Secondly, α is calculated using expression (5).

[0088] β is chosen from a finite number of candidate values, for example$\left\{ {\frac{1}{2},\frac{1}{\sqrt{2}},1,{\sqrt{2,}2}} \right\}$

[0089] and most of the calculations can be tabulated. The only componentwhich actually depends on the subband is$\frac{1}{\beta}{{\log \left( {\beta \quad W_{jk}^{\beta}} \right)}.}$

[0090] The calculation of α is hence explicit.

[0091] To avoid any confusion, it is stated here that, strictly, thesecoefficients should be called α_(jk) and β_(jk) since they are, apriori, different for each subband.

[0092] In the example above, for the subband W_(1,1) the calculation ofW_(1,1) ^(β) and of the expression (6) for values of β in the set$\left\{ {\frac{1}{2},\frac{1}{\sqrt{2}},1,{\sqrt{2,}2}} \right\}$

[0093] gives the values set out in table 1 below: TABLE 1Values  of  W_(1, 1)^(β)  and  H(p₁||G_(α(β), β))  for  the  candidate  values  of  β.

β 1/2 $1/\sqrt{2}$

1 $\sqrt{2}$

2 W_(1, 1)^(β)

1.42 1.88 3.17 7.96 38.6 H(p₁||G_(α(β), β))

2.70 2.74 2.85 3.02 3.25

[0094] The minimum is therefore reached for β={fraction (1/2)}. Thecorresponding value of α is then$\alpha = {\left( \frac{W_{1,1}^{1/2}}{2} \right) = 0.502}$

[0095] The following is therefore obtained: α_(1,1)=0.502 and β_(1,1)=½.

[0096] The parameters α and β for the other subbands are calculated inthe same way.

[0097] After having determined α and β, the relationship between rate Rand quantization step Δ and between the distortion D and thequantization step are determined. The graphs R(Δ) and D(Δ) are tabulatedand used in such a way that, for each subband, an optimum quantizationstep is obtained, that is, in such a way that the allocated rate R isdistributed in the various subbands and that the total distortion isminimal.

[0098] Prediction of the Relationship between Rate and Quantization Step

[0099] If a subband has a statistical distribution described by ageneralized Gaussian G_(α,β), the associated rate (in bits percoefficient) is noted r(α,β,Δ), for a quantization step Δ.

[0100] This rate is the entropy of the quantized subband with aquantization step Δ, which is, for example in the case of a uniformquantization:${r\left( {\alpha,\beta,\Delta} \right)} = {- {\sum\limits_{k = {- \infty}}^{+ \infty}\quad {{p_{k}\left( {\alpha,\beta,\Delta} \right)}\log_{2}{p_{k}\left( {\alpha,\beta,\Delta} \right)}}}}$

[0101] ifp_(k)(α, β, Δ) = ∫_((k − 1/2)Δ)^((k + 1/2)Δ)G_(α, β)(x)  x

[0102] The rate allocation means must know the relationship between rand (α,β,Δ). For this purpose, it is sufficient to tabulate the functionr. Since it can be verified that:${r\left( {\alpha,\beta,\Delta} \right)} = {r\left( {1,\beta,\frac{\Delta}{\alpha}} \right)}$

[0103] it is sufficient to tabulate the functions x

r(1,β,x) for all the candidate values of β (there are five in the aboveexample). The x values are also considered in a range [x_(min),x_(max)].

[0104] Note that the same type of calculations and tabulations can beperformed if a quantizer is used, having a quantization interval,centered at 0, of different size, as is often the case in the coding ofimages by wavelets.

[0105] For a subband of size n_(jk), the total rate will then be

R _(jk) =n _(jk) r(α,β,Δ).

[0106]FIG. 2 indicates tabulated values of r(1,β,x), for values of β in$\left\{ {\frac{1}{2},\frac{1}{\sqrt{2}},1,{\sqrt{2,}2}} \right\}.$

[0107] In FIG. 2, the values of x are plotted as abscissae and thevalues r of rate are plotted as ordinates.

[0108] Relationship between Distortion and Quantization Step

[0109] In the same way, the relationship between quantization step anddistortion can be modeled in a subband having a statistical distributiontending toward a generalized Gaussian. The average distortion percoefficient is noted D(α,β,Δ), for a subband of generalized Gaussianstatistical distribution G_(αβ).

[0110] This distortion is written:${\left( {\alpha,\beta,\Delta} \right)} = {\int_{- \infty}^{+ \infty}{{G_{\alpha\beta}(x)}\left( {x - {\Delta \left\lbrack \frac{x}{\Delta} \right\rbrack}} \right)^{2}\quad {x}}}$

[0111] where [x] denotes the integer that is closest to x, for the casein which the quantization operator is a uniform quantization.

[0112] Here also, the values of d(α,β,Δ) can be tabulated economically,by making use of the following homogeneity equation:

d(α,β,Δ)=α² d(1,β,Δ/α)

[0113] and it will therefore be sufficient to tabulate the function

x

d(1,β,x)

[0114] for values of β in the range of chosen candidate values, andvalues of x selected in [x_(min),x_(max)].

[0115] Here again, the total distortion for a subband is the sum ofdistortions per coefficients, and it is therefore written:

D _(jk) =n _(jk) d(α,β,Δ)

[0116] The tabulated graphs are shown in FIG. 3.

[0117] In said FIG. 3, the x values are plotted as abscissa and thedistortion as ordinate.

[0118] For a fixed α and β, the relationship between r and Δ isinvertible. This inversion is performed either by tabulation in advance,or by interpolation of the tabulated values of r(1,β,x). Similarly, therelationship between d and Δ can be inverted.

[0119] The function which associates a quantization step Δ with adistortion d is noted d⁻¹ and the function which associates thequantization step Δ with a rate r is noted r⁻¹, for fixed α and β:

Δ=d ⁻¹(α,β,d)

Δ=r ⁻¹(α,β,r)

[0120] The rate scheduling consists in cutting the set rate R intofragments which will be promptly allocated. This assignment is carriedout iteratively.

[0121] For the initialization, the starting point, for each subband, isa maximum quantization step specified by the tables of R(α,β,Δ) andD(α,β,Δ), which will be: $\Delta_{jk}{\frac{x_{\max}}{\alpha_{jk}}.}$

[0122] Rates R_(jk) and distortions D_(jk) are associated with thesequantization steps by the formulae:

R _(jk) =n _(jk) r(α_(jk),β_(jk),Δ_(jk))

D _(jk) =n _(jk) d(α_(jk),β_(jk),Δ_(jk))

[0123] The rate remaining to be allocated is therefore$R - {\sum\limits_{jk}^{\quad}\quad {R_{jk}.}}$

[0124] This rate is cut into N fragments F_(n) which can be of the samesize or of different sizes, with${F_{1} + \ldots + F_{N}} = {R - {\sum\limits_{jk}^{\quad}\quad {R_{jk}.}}}$

[0125] Above all, the maximum size of a fragment must remain small inview of the total rate setting. Each of these fragments is thenallocated iteratively, as follows:

[0126] For each subband, the potential rate is calculated which would bethe rate associated with the subband if this new rate fragment F_(n)happened to be allocated to it: R_(jk)^(*) = R_(jk) + F_(n).

[0127] The associated potential quantization steps are then calculated:$\Delta_{jk}^{*} = {r^{- 1}\left( {\alpha_{jk},\beta_{jk},\frac{R_{jk}^{*}}{n_{jk}}} \right)}$

[0128] then the new associated distortions:D_(jk)^(*) = n_(jk)d(α_(jk), β_(jk), Δ_(jk)^(*))

[0129] For each subband j,k it can be estimated what the reduction indistortion would be if the rate fragment F were to be allocated to it.This reduction would be: D_(jk) − D_(jk)^(*)

[0130] The “best” subband j,k, that is the one for which the reductionin distortion is strongest, is allocated the fragment:R_(jk) ← R_(jk)^(*) D_(jk) ← D_(jk)^(*) Δ_(jk) ← Δ_(jk)^(*)

[0131] The same rate allocation is reiterated for the next fragmentsF_(n+1), F_(n+2), until the rate to be allocated is exhausted.

[0132] Table 2 below indicates, for the abovementioned example, whichare the values of α and β retained. The coding of the low-pass subbandW_(3,0) is performed in DPCM (Difference Pulse Code Modulation), and thestatistics indicated in the table are therefore statistics ofW_(3,0)[n_(k),m_(k)]−W_(3,0)[n_(k−1),m_(k−1)], where the numbering ofpairs

k

(n _(k) ,m _(k))

[0133] indicates the direction chosen for the low-pass coefficientsduring the coding. TABLE 2 Laplacian parameters for a test image j k αjk β_(jk) D_(jk) R_(jk) 1 1 0.50 1/2 50 444 1 2 0.35 1/2 35 444 1 3 0.561{square root}2 56 0 2 1 1.50 1/2 150 111 2 2 1.20 1/2 120 111 2 3 0.751/2 75 111 3 1 3.60 1/2 360 27.8 3 2 3.40 1/2 340 27.8 3 3 2.10 1/2 21027.8 3 0 11 1/2 1100 27.8

[0134] The total rate allocated is therefore at least the sum of therates R_(jk) above, that is 1332 bits, therefore 0.005 bits per pixel.The setting is of 1 bit per pixel, that is a total budget of R=262144bits, the image being of size 512×512.

[0135] The remaining rate to be allocated of 260812 bits is divided intoparts that are not necessarily equal, but of sizes that are all lessthan a fixed limit. The first rate fragment to be allocated is forexample of 384 bits.

[0136] The tables are used to calculate, for each subband, what thedistortion associated with each subband will be if a budget of 384additional bits were to be attributed to it. The result of thesecalculations is set out in table 3.

[0137] For example, for the subband W_(1,2), the total rate R_(1,2),would change from 444 bits to 828 bits, and the distortion D_(1,2),would change from 0.93×10⁶ to 0.89×10⁶ and would therefore be reduced by0.03×10⁶. TABLE 3 Simulations for allocating the first fragment. Thedistortions are to be multiplied by 10⁶. j k R_(jk) Δ_(jk) D_(jk)R*_(jk) Δ*_(jk) D*_(jk) D_(jk)-D*_(jk) 1 1 444 50 1.86 828 43 1.79 0.071 2 444 35 0.93 828 30 0.89 0.04 1 3 0 35 0.19 384 12 0.18 0.01 2 1 111150 4.01 495 98 3.56 0.45 2 2 111 120 2.71 495 80 2.40 0.31 2 3 111 751.03 495 49 0.91 0.12 3 1 27.8 360 5.85 411 154 4.11 1.74 3 2 27.8 3405.38 411 148 3.78 1.60 3 3 27.8 210 1.96 411 89 1.37 0.59 3 0 27.8 11005.24 411 461 3.68 1.56

[0138] In table 3, it is seen that the strongest reduction is obtainedby allocating the 384 bits to the subband W_(3,1). Therefor Δ_(3,1),R_(3,1) and D_(3,1) are updated and the process is continued by theallocation of the next fragment. When all the fragments have beenallocated, quantization setting Δ_(jk), and prediction on the associatedrate R_(jk) and distortion D_(j) for each subband, are thereforeobtained.

[0139] The table finally obtained for the image is given in table 4below: TABLE 4 Example of settings of quantization and associated ratesby the scheduler. The rates are given in bits per coefticient. j k Δ jkR_(jk) 1 1 6.1 0.92 1 2 6.5 0.53 1 3 8.4 0.03 2 1 5.7 2.5 2 2 5.7 2.1 23 5.8 1.4 3 0 6.3 3.8 3 1 5.8 3.7 3 2 6 2.9 3 3 5.9 5.5

[0140] The allocation of bits can be performed in open-loop mode or,preferably, in closed-loop mode with a device of the type illustrated inFIG. 4.

[0141] The rate regulation obtained using the device illustrated in FIG.4 is to the nearest bit. This device is based on the one described inthe French patent filed on Mar. 18, 1999 under the number 99/03371.

[0142] In this patent, a technique is described for the acquisition ofimages of the Earth by a moving observation satellite (“Push Broom”mode) in which the images are compressed and transmitted to the groundvia a transmission channel which imposes a constant rate. To achievethis objective, an optimum allocation of bits is performed, in real timeor in slightly deferred time, in order to code the subband coefficients,and then the allocation errors and variations in rate at the output ofthe entropy coder are corrected.

[0143] This device includes a compression means 20, and the image datais applied to the input 22 of said compression means 20. The compressionmeans 20 includes a wavelet transformation unit 24 supplying data to asubband quantizer which delivers data to a coder 28 the output of whichforms the output of the compression means 20. This output of thecompression means 20 is linked to a regulation buffer memory 30supplying, at its output, compressed data according to a rate R_(c).

[0144] The coder 28 also supplies a number N_(p) of bits produced whichis applied to an input 33 of a control unit 32 having another input forthe set rate R_(c) and an output applied to a first input of a bitallocation unit 34. This latter unit 34 has another input 36 to whichthe set rate R_(c) is applied and an input 38 to which the output datafrom the transformation unit 24 is applied.

[0145] The output of the unit 34 is applied to an input 40 of thesubband quantizer 26.

[0146] The method and device in accordance with the invention enable thecost of coding the image to be reduced and ensure a rapid rateregulation since, unlike in the prior art, no iterative optimization(Lagrangian for example) is performed. The calculations are simple.Thus, the method in accordance with the invention is well suited for allthe applications and, in particular, for space applications in which theresources are necessarily limited.

[0147] It is to be noted that, in the present description, the term“image” must be understood as being in the sense of an image of at leastone dimension, that is to say that the invention extends to data ingeneral.

There is claimed:
 1. A method for compressing data, in particularimages, by transform, in which method this data is projected onto a baseof localized orthogonal or biorthogonal functions, and which method, toquantize each of the localized functions with a quantization step thatenables an overall set rate R_(c) to be satisfied, includes thefollowing steps: a) a probability density model of coefficients in theform of a generalized Gaussian is associated with each subband, b) theparameters α and β of this density model are estimated, while minimizingthe relative entropy, or Kullback-Leibler distance, between this modeland the empirical distribution of coefficients of each subband, and c)from this model, for each subband, an optimum quantization step isdetermined such that the rate allocated is distributed in the varioussubbands and such that the total distortion is minimal.
 2. The methodclaimed in claim 1, wherein, for each subband, the graphs of rate R anddistortion D are deduced, from the parameters α and β, as a function ofthe quantization step and said graphs are tabulated to determine saidoptimum quantization step.
 3. The method claimed in claim 1, whereinsaid functions are wavelets.
 4. The method claimed in claim 3, wherein,to determine said parameter β of the generalized Gaussian associatedwith each subband, the following expression is minimized:${H\left( p_{1}||G_{{\alpha {(\beta)}},\beta} \right)} = {\frac{1}{\beta} + {\frac{1}{\beta}{\log \left( {\beta \quad W_{jk}^{\beta}} \right)}} + {\log \left( \frac{2{\Gamma \left( {1/\beta} \right)}}{\beta} \right)}}$

in which formula the first term represents said relative entropy,W_(jk)^(β)

has the value:$W_{jk}^{\beta} = {\frac{1}{n_{jk}}{\sum\limits_{n,m}^{\quad}{{W_{jk}\left\lbrack {n,m} \right\rbrack}}^{\beta}}}$

W_(jk)[n,m] being a coefficient of a subband, and n_(jk) being thenumber of coefficients in the subband of index j,k.
 5. The methodclaimed in claim 4, wherein said parameter α of the generalized Gaussianfor the corresponding subband j,k is determined by the followingformula: $\alpha = \sqrt[\beta]{\beta \quad W_{jk}^{\beta}}$


6. The method claimed in claim 3, wherein the tabulation of thedistortion values is carried out for a sequence of values of theparameter β.
 7. The method claimed in claim 3, wherein the tabulation ofthe rates R is carried out for a sequence of values of the parameter β.8. The method claimed in claim 6, wherein the sequence of values of theparameters β is$\left\{ {\frac{1}{2},\frac{1}{\sqrt{2}},1,\sqrt{2},2} \right\}.$


9. The method claimed in claim 1, wherein the bit budget is distributedto each of the subbands according to their ability to reduce thedistortion of the compressed image.
 10. The method claimed in claim 9,wherein a bit budget corresponding to the highest tabulated quantizationstep is assigned to each subband and wherein the remaining bit budget isthen cut into individual parts that are gradually allocated to thelocalized functions having the greatest ability to make the totaldistortion decrease, this operation being repeated until the bit budgetis exhausted.
 11. The method claimed in claim 1, wherein the localizedfunctions and the quantization step for each subband and the parametersof the density model are coded using a lossless entropy coder.